MathDB

x^2 + px+q = 0, x^2 +rx+s = 0 => pr = (q+1)(s+1) , p(q+1)s = r(s+1)q

Source: Czech and Slovak MO, III A, 2005 p5

January 12, 2020

quadraticsquadratic trinomialtrinomialalgebra

Problem Statement

Let

p,q, r, s

be real numbers with

q \ne -1

and

s \ne -1

. Prove that the quadratic equations

x^2 + px+q = 0

and

x^2 +rx+s = 0

have a common root, while their other roots are inverse of each other, if and only if

pr = (q+1)(s+1)

and

p(q+1)s = r(s+1)q

. (A double root is counted twice.)